Practical Ship Hydrodynamics Episode 11 pdf

Ship manoeuvring 191 average between V A and V 1 : r 1 D r 0  1 2  1 C V A V 1  Here r 0 is half the propeller diameter D. Normally the rudder is in a position where the slipstream contraction is not yet completed. The slipstream radius and axial velocity may be approximated by (S ¨ oding (1982)): r x D r 0 Ð 0.14r 1 /r 0  3 C r 1 /r 0 Ð x/r 0  1.5 0.14r 1 /r 0  3 C x/r 0  1.5 and: V x D V 1 Ð  r 1 r  2 Here x is the distance of the respective position behind the propeller plane. To determine rudder force and moment, it is recommended to use the position of the centre of gravity of the rudder area within the propeller slipstream. The above expression for r x is an approximation of a potential-flow calcu- lation. Compared to the potential flow result, the slipstream will increase in diameter with increasing the distance x from the propeller plane due to turbu- lent mixing with the surrounding fluid. This may be approximated (S ¨ oding (1986)) by adding: r D 0.15x Ð V x  V A V x C V A to the potential slipstream radius and correcting the slipstream speed according to the momentum theorem: V corr D V x  V A   r r C r  2 C V A The results of applying this procedure are shown in Fig. 5.19. V corr is the mean value of the axial speed component over the slipstream cross-section. The rudder generates a lift force by deflecting the water flow up to consider- able lateral distances from the rudder. Therefore the finite lateral extent of the propeller slipstream diminishes the rudder lift compared to a uniform inflow velocity. This is approximated (S ¨ oding (1982)) (based on two-dimensional potential flow computations for small angles of attack) by multiplying the rudder lift determined from the velocity within the rudder plane by the correc- tion factor  determined from:  D  V A V corr  f with f D 2 Ð  2 2 C d/c  8 Here V A is the speed outside of the propeller slipstream laterally from the rudder. d is the half-width of the slipstream. For practical applications, it is recommended to transform the circular cross-section (radius r C r)ofthe 192 Practical Ship Hydrodynamics 3.0 V corr / V A 2.5 2.0 1.7 1.5 1.4 1.3 1.2 1.1 1.0 1.0 0.9 0.8 ( r + D r ) / r 0 0 X / D 7.5 C Th C Th 10.0 5.0 4.0 3.0 2.5 2.0 1.5 1.0 0.75 0.5 0.5 2 5 20 50 0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 X / D Figure 5.19 Mean axial slipstream speed as a function of propeller approach speed V A and slipstream radius r C r/r 0 due to potential flow and turbulent mixing at different positions x/D behind the propeller propeller slipstream to a quadratic one (edge length 2d)ofequalarea.This leads to the relation: d D   4 r C r D 0.886 Ðr Cr The inflow velocity in the rudder plane varies along the rudder height due to the wake distribution and the propeller slipstream. The effect of this variation may be approximated by using the mean squared velocity: V 2 D 1 A R  b 0 V 2 Ð c Ð dz for the determination of the rudder lift. However, lifting-surface calculations show that, compared to a uniform distribution, the lift coefficient (defined with reference to V 2 ) is some 5% higher for rudders extending downward beyond the lower edge of the propeller slipstream (Fig. 5.20). Hence it is recommended to extend the rudder as far to the base line of the ship as possible. Ship manoeuvring 193 1.05 1.00 0.95 C L ( V corr = V A ) C L ( V corr ≠ V A ) b b 0.15 . b V A V A V A 0.85 . b 0.30 . b V corr V corr V corr / V A 1234567 L = 0.5 L = 2.5 L = 2.5 L = 0.5 Figure 5.20 Lift coefficients as a function of the vertical distribution of flow and the aspect ratio  A simple global correction for the lift force of a rudder behind a propeller (to be added to the lift computed by the usual empirical formulae for rudders in free stream) is (S ¨ oding (1998a, b)): L D T Ð  1 C 1  1 C C Th  Ð sin υ The additional drag (or decrease in propeller thrust) is: D D T Ð  1 C 1  1 C C Th  Ð 1 cos υ 5.4.5 Interaction of rudder and ship hull If the hull above the rudder is immersed, it suppresses the flow from the pressure to the suction side around the upper edge of the rudder. This has effects similar to an increase of the rudder aspect ratio : ž It decreases the induced drag. ž It increases the slope of the lift curve versus angle of attack ˛. ž It hardly influences the maximum lift at the stall angle ˛ s . The magnitude of this effect depends on the size of the gap between the upper edge of the rudder and the hull. For very small gaps, the aspect ratio  eff is theoretically twice the nominal value, in practice  eff ³ 1.6 Ð geom .Toclose 194 Practical Ship Hydrodynamics the gap between hull and rudder at least for small rudder angles υ – and thus increasing the rudder effectiveness – a fixed fin above the rudder is advanta- geous for small–rudder angles. If the hull above the rudder is not immersed or if the rudder intersects the water surface, the free surface may also increase somewhat the effective aspect ratio  eff . However, this effect decreases with increasing ship speed and may turn to the opposite at higher speed by rudder ventilation drawn from the surface along the suction side of the rudder. To decrease rudder ventilation, a broad stern shape sufficiently immersed into the water especially above the front part of the rudder is advantageous. The wake of the hull decreases the inflow velocity to the rudder and increases the propeller load. Differences in wake and propeller load between model and ship are the main cause of scale effects in model manoeuvring experiments. Whereas the wake due to hull surface friction will be similar at the rudder and at the propeller, the potential wake – at least for small Froude numbers, i.e. without influence of the free surface – is nearly zero at the rudder, but typically amounts to 10% to 25% of the ship’s speed at the propeller of usual single- screw ships. It amounts nearly to the thrust deduction fraction t. Thus the flow outside of the propeller slipstream is accelerated between the propeller and the rudder by about t ÐV. This causes a pressure drop which also accelerates the propeller slipstream to approximately: V x D V 2 corr C t Ð V 2 /V corr The corresponding slipstream contraction is: r x D r C r Ð  V corr /V x For non-zero rudder angle and forward ship speed, an interaction between the flow around rudder and hull occurs which decreases the lift force at the rudder; however, an additional transverse force of equal direction is generated at the aftbody. Compared to the rudder lift without hull interaction, the total transverse force is increased by the factor 1 Ca H Ð a H may be approximated (S ¨ oding (1982)): a H D 1 1 C 4.9 Ð e/T C 3 Ðc/T 2 Here T is the draft of the ship, e the mean distance between the front edge of the rudder and the aft end of the hull, and c the mean rudder chord length. Compared to the free-running rudder, the centre of effort of the total transverse force is shifted forward by approximately: x D 0.3T e/T C 0.46 Potential flow computations show that x mayincreasetouphalftheship’s length in shallow water if the gap length e between rudder and hull tends to zero, as may be the case for twin-screw ships with a centre rudder. This would decrease the ship’s turning ability on shallow water. For a non-zero drift velocity v (positive to starboard, measured amidships) and a non-zero yaw rate r (positive clockwise if seen from above) of the ship, the hull in front of the rudder influences the flow direction at the rudder position. Without Ship manoeuvring 195 hull influence, the transverse flow velocity v relative to the hull at the rudder position x R is: v R Dv C x R Ð r where x R is the distance between rudder and midship section, negative for stern rudders. However, experiments of Kose (1982) with a freely rotating, unbalanced rudder behind a ship model without propeller indicated a mean transverse velocity at the rudder’s position of only: v R D0.36 Ðv C 0.66 Ðx R Ð r From the rudder angle υ (positive to port side), the mean longitudinal flow speed V x (positive backward) and the mean transverse flow speed v R at the rudder position, the angle of attack follows: ˛ D υ C arctan v R V corr 5.4.6 Rudder cavitation Rudder cavitation may occur even at small rudder angles for ship speed’s exceeding 22 knots with rudder(s) in the propeller slipstream and: P D 2 /4 > 700 kW/m 2 Here P is the delivered power, D the propeller diameter. Rudder cavitation – as with propeller cavitation – is caused by water evapo- ration where at points of high flow velocity the pressure locally drops below the vapour pressure of the water. Cavitation erosion (loss of material by mechan- ical action) occurs when small bubbles filled with vapour collapse on or near to the surface of the body. During the collapse a microscopic high-velocity jet forms, driven by surface tension and directed onto the body surface. It causes small cracks and erosion, which in seawater may be magnified by corrosion (galvanic loss of material). Paint systems, rubber coatings, enamel etc. offer no substantial resistance to cavitation, but austenitic steel and some types of bronze seem to retard the erosion compared to the mild steel normally used for rudders. The cavitation number  (Fig. 5.21) is a non-dimensional characteristic value for studying cavitation problems in model experiments:  D p  p v  2 V 2 where p is the pressure in undisturbed flow, i.e. atmospheric pressure plus hydrostatic pressure, p v vaporization pressure, V ship speed,  density of water. There are different types of rudder cavitation: 1. Bubble cavitation on the rudder side plating For large rudder angles, cavitation is unavoidable in ships of more than about 10 knots. It will decrease the rudder lift substantially if the cavitation 196 Practical Ship Hydrodynamics h = 8 m 10 m 12 m 14 m 16 m 18 m 20 m 8 7 6 5 4 3 2 1 0 16 20 24 28 32 36 V (kn) h = 2 m h = 0 m h = 6 m 1m 5 m 4 m 3 m d = P − P V r . V 2 / 2 Figure 5.21 Cavitation numbers  as a function of the ship speed V with the submersion h (depth below water surface) as parameter causes a complete separation of flow from the suction side. Otherwise its influence on rudder forces is small (Kracht (1987)). Cavitation erosion is of interest only if it occurs within the range of rudder angles υ Dš5 ° used for course keeping. Evaluation of model experiments shows that the onset of cavitation is indeed observed if the pressure determined by potential-flow theory is smaller than the water vaporization pressure p v . p v lies typically between 1% and 3% of the atmospheric pressure. It may therefore (not in model tests, but for full-scale ships) simply be taken as zero. Thus, to test for blade side cavitation in the design stage of ships, one may proceed as follows: – Determine the slipstream radius r C r and the inflow speed to the rudder V corr from Fig. 5.19 or the corresponding formulae at about 80% of the propeller tip radius above and below the propeller axis. – Correct these values to obtain V x and r x by (see above): V x D V 2 corr C t Ð V 2 /V corr r x D r C r Ð  V corr /V x – Because of non-uniform distribution of the slipstream velocity, add 12% of V to obtain the maximum axial speed at the rudder: V max D V x C 0.12 Ð V corr  V A  Ship manoeuvring 197 – Estimate the inflow angle ˛ to the rudder due to the rotation of the propeller slipstream by ˛ D arctan  4.3 Ð K Q J 2 Ð  1  w 1  w local Ð V A V max  w is the mean wake number and w local the wake number at the respective position. The equation is derived from the momentum theorem with an empirical correction for the local wake. It is meant to apply about 0.7 to 1.0 propeller diameter behind the propeller plane. The position relevant to the pressure distribution is about 1/2 chord length behind the leading edge of the rudder. –Addυ D 3 ° D 0.052 rad as an allowance for steering rudder angles. – Determine the maximum local lift coefficient C Ll max from Fig. 5.22, where ˛ Cυ are to be measured in radians. c is the chord length of the rudder at the respective height, r x the propeller slipstream radius (see above): r x D r o Ð  1 2  1 C V A V 1  2.4 2.2 2.0 1.8 1.6 1.4 0.8 1.0 1.2 1.4 1.6 1.8 r x C Ll max ⋅ C (a + d) ⋅ r x Figure 5.22 Diagram for determining the local values, of maximum lift coefficient C Ll max Figure 5.22 is based on lifting-line calculations of a rudder in the propeller slipstream. It takes into account the dependence of the local lift coefficient on the vertical variation of inflow velocity and direction. – Determine the extreme negative non-dimensional pressure on the suction side depending on profile and local lift coefficient C Ll max . For this we use Fig. 5.23 derived from potential flow calculations. –Addtop dyn (negative) the static pressure p stat D 103 kPa C Ð g Ðh. h is the distance between the respective point on the rudder and the water surface, e.g. 80% of the propeller radius above the propeller axis. 198 Practical Ship Hydrodynamics 2.5 2.0 1.5 1.0 0.5 0 0.25 0.50 C Ll 0.75 IFS 61−TR 25 IFS 62−TR 25 NACA−0015 IFS 58−TR 15 NACA−0024 NACA−0021 NACA−0018 NACA 64 2 −015 NACA 64 3 −018 NACA 64 4 −021 HSVA−MP 73−20 P dyn (ρ ⋅ V 2 max ) / 2 HSVA−MP 71−20 Figure 5.23 Extreme negative dynamic pressure of the suction side as a function of the local lift coefficient C Ll and the profile If the resulting minimum pressure on the suction side is negative or slightly positive (less than 3 kPa), the side plating of the rudder is prone to cavita- tion. For a right-turning propeller (turning clockwise looking forward) the cavitation will occur: – on the starboard side in the upper part of the rudder relative to the propeller axis – on the port side in the lower part of the rudder relative to the propeller axis Brix (1993), pp. 91–92, gives an example for such a computation. Measures to decrease rudder side cavitation follow from the above prediction method: – Use profiles with small p dyn at the respective local lift coefficient. These profiles have their maximum thickness at approximately 40% behind the leading edge. – Use profiles with an inclined (relative to the mean rudder plane) or curved mean line to decrease the angle of attack (Brix et al. (1971)). For a right- turning propeller, the rudder nose should be on the port side above the propeller axis, on the starboard side below it. 2. Rudder sole cavitation Due to the pressure difference between both sides of the rudder caused, e.g., by the rotation of the propeller slipstream, a flow component around the rudder sole from the pressure to the suction side occurs. It causes a Ship manoeuvring 199 rudder tip vortex (similar to propeller tip vortices) which may be filled by a cavitation tube. This may cause damage if it attaches to the side of the rudder. However, conditions for this are not clear at present. If the rudder has a sharp corner at the front lower edge, even for vanishing angles of attack the flow cannot follow the sharp bend from the leading edge to the base plate, causing cavitation in the front part of the rudder sole. As a precaution the base plate is bent upward at its front end (Brix et al. (1971)). This lowers the cavitation number below which sole cavitation occurs (Fig. 5.24). For high ship speeds exceeding, e.g., 26 knots cavitation has still been reported. However, it is expected that a further improvement could be obtained by using a smoothly rounded lower face or a baffle plate at the lower front end (Kracht (1987)). No difficulties have been reported at the rudder top plate due to the much lower inflow velocity. 0° 5° 10° 15° 20° 25° ∝ 0 1 2 3 δ = P − P V r ⋅ V 2 /2 Flat rudder sole Rounded leading edge Figure 5.24 Cavitation number  below which rudder sole cavitation occurs as a function of theangleofattack˛ and the rudder sole construction 3. Propeller tip vortex cavitation Every propeller causes tip vortices. These are regions of low pressure, often filled with cavitation tubes. Behind the propeller they form spirals which are intersected by the rudder. Therefore, cavitation erosion frequently occurs at the rudder at the upper and sometimes lower slipstream boundaries, mainly (for right-turning propellers) on the upper starboard side of the rudder. This problem is not confined to high-speed ships. The best means to reduce these effects is to decrease gradually the propeller loading to the blade tips by appropriately reduced pitch, and to use a high propeller skew. These methods also reduce propeller-induced vibrations. 4. Propeller hub cavitation Behind the propeller hub a vortex is formed which is often filled by a cavitation tube. However, it seems to cause fewer problems at the rudder 200 Practical Ship Hydrodynamics than the tip vortices, possibly due to the lower axial velocity behind the propeller hub. 5. Cavitation at surface irregularities Surface irregularities disturbing the smooth flow cause high flow velocities at convex surfaces and edges, correspondingly low pressures and frequently cavitation erosion. At the rudder, such irregularities may be zinc anodes and shaft couplings. It is reported that also behind scoops, propeller bossings etc. cavitation erosion occurred, possibly due to increased turbulence of the flow. Gaps between the horn and the rudder blade in semi-balanced rudders are especially prone to cavitation, leading to erosion of structurally important parts of the rudder. For horizontal and vertical gaps (also in flap rudders) the rounding of edges of the part behind the gap is recommended. 5.4.7 Rudder design There are no regulations for the rating of the rudder area. The known recom- mendations give the rudder area as a percentage of the underwater lateral area L ÐT. Det Norske Veritas recommends: A R L ÐT ½ 0.01 Ð  1 C 25  B L  2  This gives a rudder area of approximately 1.5% of the underwater lateral area for ships of usual width; for unusually broad ships (large mass, low yaw stability) a somewhat larger value is given. This corresponds to typical rudder designs and can serve as a starting point for further analyses of the steering qualities of a ship. Recommended minimum criteria for the steering qualities of a ship are: ž Non-dimensional initial turning time in Z 20 ° /10 ° manoeuvres: t 0 a D 1 C 1.73F n . ž Non-dimensional yaw checking time in Z 20 ° /10 ° manoeuvres: t 0 s D 0.78 C 2.12F n . ž The rudder should be able to keep the ship on a straight course with a rudder angle of maximum 20 ° for wind from arbitrary direction and v w /V D 5. v w is the wind speed, V the ship speed. ž The ship must be able to achieve a turning circle of less than 5 ÐL at the same v w /V for maximum rudder angle. The criteria for initial turning time and yaw checking time were derived by Brix using regression analysis for 20 ° /10 ° zigzag test results for many ships (Fig. 5.8). The time criteria are critical for large ships (bulkers, tankers), while the wind criteria are critical for ships with a large lateral area above the water (ferries, combatants, container ships). An additional criterion concerning yaw stability would make sense, but this would be difficult to check computationally. The rudder design can be checked against the above criteria using computa- tions (less accurate) or model tests (more expensive). A third option would be the systematically varied computations of Wagner, described in Brix (1993), pp. 95–102. This approach yields a coefficient C Yυ for rudder effectiveness which inherently meets the above criteria. The method described in Brix [...]... (www.bh.com/companions/0750648 511) 1 A motor yacht of 10 000 kg displacement is equipped with a 1 m2 profile rudder with  D 1.2 The yacht is a twin-screw ship with central rudder The rudder lies outside the propeller slipstream The yacht has a speed of 13.33 m/s For the central position of the rudder we can assume a velocity of 0.75 ship speed due to the wake The ‘glide ratio’ (ratio of propeller thrust to ship weight)... manoeuvre the ship by turning the jets a maximum of 35° , just as previously the maximum rudder angle was 35° The speed may be assumed to be unaffected by the conversion Will the yacht react faster or slower after its conversion? Why? 2 A tanker of 250 000 t displacement sails at 15 knots at a delivered power PD D 15 000 kW The overall efficiency is ÁD RÐU D 0.7 PD 204 Practical Ship Hydrodynamics. . .Ship manoeuvring 201 (1993) uses design diagrams For computer calculations, empirical formulae also derived by Wagner exist 5.4.8 CFD for rudder flows and conclusions for rudder design The determination of forces on the rudder is important for practical design purposes: ž The transverse force is needed to evaluate the manoeuvrability of ships already in the design stage... linearly increasing with ˛ corresponding to: dCL D2 d˛ In three-dimensional flow, the lift gradient is decreased by a reduction factor r  which is well approximated by: r  D  C 0.7  C 1.7 2 202 Practical Ship Hydrodynamics Except for , details of the rudder shape in side view (e.g rectangular or trapezoidal) have hardly any influence on dCL /d˛ However, the profile thickness and shape have some influence... jT/Rj D 0.945 3 (a) A ship lays rudder according to sine function over time alternating between port and starboard with amplitude 10° and 2 minutes period The ship performs course changes of š20° The maximum course deviation to port occurs 45 seconds after the maximum rudder angle to port has been reached Determine from these data the parameters of the Nomoto equation: TR C P D Kυ The ship is yaw stable... and the delay between maximum rudder angle and maximum course deviation? Hint: For constant rudder angle, the Nomoto equation has the solution: P D ae˛t C b 4 A ship follows the ‘Norrbin’ equation: TR C P C˛P3 D Kυ Ship manoeuvring 205 The ship performs a pull-out manoeuvre and the following curve for P is recorded: • 0.020 y (rad/s) 0.010 0.007 t (s) 10 20 (a) Determine the constants T and ˛! (b)... potential are transformed from the local x-y-z system to a global x-y-z system In two dimensions, we limit ourselves to x and z as coordinates, as these are the typical coordinates for a 207 208 Practical Ship Hydrodynamics strip in a strip method n D nx , nz is the outward unit normal in global coorE dinates, coinciding with the local z vector E and E are unit tangential vectors, t s coinciding with... be physically interpreted as a source of water which constantly pours water flowing radially in all directions is the strength of this source For negative , the element acts like a sink with 210 Practical Ship Hydrodynamics Figure 6.1 Effect of a point source water flowing from all directions into the centre Figure 6.1 illustrates the effect of the element Higher derivatives are: D zz D xz xx 1 2 r2 D... occurs 45 seconds after the maximum rudder angle to port has been reached Determine from these data the parameters of the Nomoto equation: TR C P D Kυ The ship is yaw stable for K > 0 Is the ship yaw stable? (b) The ship speed is reduced to 50% of the value in (a) The rudder action is the same as in (a) How large is the amplitude of course changes and what is the delay between maximum rudder angle and... ratio produces somewhat smaller CL,max ž Small  yield large stall angles (They also yield small dCL /d˛, hence little change in the maximum CL ) ž The taper ratio of the rudder has practically no influence on the maximum CL Ship manoeuvring 203 ž Profiles with concave sides produce larger CL,max than those with flat or convex sides Three-dimensional RANSE computation give slightly lower maximum CL than . half-width of the slipstream. For practical applications, it is recommended to transform the circular cross-section (radius r C r)ofthe 192 Practical Ship Hydrodynamics 3.0 V corr / V A 2.5 2.0 1.7 1.5 1.4 1.3 1.2 1.1 1.0 1.0 0.9 0.8 ( r .  eff is theoretically twice the nominal value, in practice  eff ³ 1.6 Ð geom .Toclose 194 Practical Ship Hydrodynamics the gap between hull and rudder at least for small rudder angles υ – and thus increasing. mayincreasetouphalftheship’s length in shallow water if the gap length e between rudder and hull tends to zero, as may be the case for twin-screw ships with a centre rudder. This would decrease the ship s turning